论文标题
空间均匀玻尔兹曼方程的傅里叶 - 加盖尔光谱法的新稳定性和收敛证明
A new stability and convergence proof of the Fourier-Galerkin spectral method for the spatially homogeneous Boltzmann equation
论文作者
论文摘要
玻尔兹曼方程的数值近似是一个具有挑战性的问题，由于其高维，非本地和非线性碰撞积分。在过去的十年中，傅立叶 - 加盖尔金光谱方法已成为解决玻尔兹曼方程的流行确定性方法，它以其高精度和被快速傅立叶变换进一步加速的潜力所表现出来。尽管其实际成功，但该方法的稳定性直到最近才由Filbet，F。＆Mouhot，C。证明。阿米尔。数学。 Soc。$ 363，没有。 4（2011）：1947-1980。]通过利用碰撞操作员的“扩散”属性。在这项工作中，我们基于仔细的$ l^2 $估计解决方案的负部分提供了新的证明。我们还讨论了结果对各种初始数据的适用性，包括连续和不连续的功能。
Numerical approximation of the Boltzmann equation is a challenging problem due to its high-dimensional, nonlocal, and nonlinear collision integral. Over the past decade, the Fourier-Galerkin spectral method has become a popular deterministic method for solving the Boltzmann equation, manifested by its high accuracy and potential of being further accelerated by the fast Fourier transform. Albeit its practical success, the stability of the method is only recently proved by Filbet, F. & Mouhot, C. in [$ Trans. Amer. Math. Soc.$ 363, no. 4 (2011): 1947-1980.] by utilizing the "spreading" property of the collision operator. In this work, we provide a new proof based on a careful $L^2$ estimate of the negative part of the solution. We also discuss the applicability of the result to various initial data, including both continuous and discontinuous functions.
